Proof-Carrying Real and Complex No-Boundary Saddles in Einstein–Scalar Gravity Finite-amplitude stability, certified complex continuation, and proof-carrying Picard–Lefschetz geometry

We study a symmetric Einstein–scalar benchmark linking real instanton stability, complexno-boundary continuation, and fixed-lapse Picard–Lefschetz geometry. A proof-carryingconstruction establishes the finite real branch, its physical Morse structure, a unique de-Sitter-connected complex saddle branch, and the Hartle–Hawking fixed-lapse thimble fromits local Morse germ to the small-lapse contour crossing. The resulting saddle-specificoriented intersection number is nHH = +1. For the competing saddles, the analysis isolates anonremovable singular-end mechanism on the inward L1 arm; the final outward cap-sensitivityenclosure and the L2, L3 classifications remain open before a complete all-saddles contourdecomposition is claimed.